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Discrete Mathematics with Graph Theory (3rd Edition) 278

Discrete Mathematics with Graph Theory (3rd Edition) 278 -...

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276 Solutions to Exercises contains VI. Since there is an edge between Vi and Vi+I for i = 1,2, ... , n - 1, it follows that Vb V3, V5,··. are in ODD and that V2, V4, V6, .•. are in EVEN. Now Vi is in EVEN; otherwise, it is in ODD along with Vb an impossibility because there is an edge between Vi and VI. So EVEN and ODD have the same number of elements; m = n = !. (b) The graph is bipartite, as shown, and a subgraph of IC 3 ,4. Since IC3,4 is not Hamiltonian, neither is any subgraph. w B w W B Exercises 10.3 0 1 0 1 0 0 [ ~ 1 0 0 H 1 0 1 1 0 0 0 1 0 1. In Fig. 10.2, [BB] A = 0 1 0 1 1 1 In Fig. 10.4, A = 1 0 1 1 1 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 1 2. The adjacency matrix of IC n is 1 1 0 1 1 1 1 ... 0 0 0 1 1 If we order the vertices of ICm,n so that the bipartition 0 0 1 1 sets are
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Unformatted text preview: sets are VI, . .. , vm and vm+b . .. , Vm+ n ' the 1 1 0 0 adjacency matrix is 1 1 0 0 3. (a) [BB] The (3,5) entry of A3 is the number of walks of length 3 from C to E and, hence, equals 5. The (2,2) entry of A3 is the number of walks oflength 3 from B to itself, hence, equals 2. (b) The (3,5) entry of A3 equals 2. The (2,2) entry of A3 equals 4. 4. [BB] Each 1 represents an edge. Each edge ViVj contributes two l's to the matrix, in positions (i,j) and (j, i). The number of l's is twice the number of edges. 5. The ith entry on the diagonal of A3 is the number of walks of length 3 from Vi to itself. However, a closed walk of length 3 in a graph must give a triangle. Every triangle can be walked in exactly two different directions. Hence, the number of walks of length 3 is twice the number of triangles....
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